Variance - Population Variance and Sample Variance

Population Variance and Sample Variance

In general, the population variance of a finite population of size N is given by

where

is the population mean, and

\begin{align} \sum_{i=1}^N \left(x_i - \mu \right)^2 &= \sum_{i=1}^N \left(x_i^2 - 2 x_i \mu + \mu^2 \right) \\ &= \sum_{i=1}^N \left(x_i^2 + \mu^2 \right) - 2 \mu \sum_{i=1}^N x_i \\ &= \sum_{i=1}^N \left(x_i^2 + \mu^2 \right) - 2 N \mu^2 \\ &= \sum_{i=1}^N \left(x_i^2 + \mu^2 - 2 \mu^2 \right) \\ &= \sum_{i=1}^N \left(x_i^2 - \mu^2 \right) \end{align}

In many practical situations, the true variance of a population is not known a priori and must be computed somehow. When dealing with extremely large populations, it is not possible to count every object in the population.

A common task is to estimate the variance of a population from a sample. We take a sample with replacement of n values y1, ..., yn from the population, where n < N, and estimate the variance on the basis of this sample. There are several good estimators. Two of them are well known:

and
\begin{align} s^2 & = \frac{1}{n-1} \sum_{i=1}^n\left(y_i - \overline{y} \right)^ 2 \\ & = \frac{1}{n-1}\sum_{i=1}^n\left(y_i^2-\overline{y}^2\right) \\ & = \frac{1}{n-1}\sum_{i=1}^n y_i^2-\frac{n}{n-1}\overline{y}^2 \end{align}

The first estimator, also known as the second central moment, is called the biased sample variance. The second estimator is called the unbiased sample variance. Either estimator may be simply referred to as the sample variance when the version can be determined by context. Here, denotes the sample mean:

The two estimators only differ slightly as can be seen, and for larger values of the sample size n the difference is negligible. While the first one may be seen as the variance of the sample considered as a population, the second one is the unbiased estimator of the population variance, meaning that its expected value E is equal to the true variance of the sampled random variable; the use of the term n − 1 is called Bessel's correction. In particular,

\operatorname{E} = \sigma^2,

while, in contrast,

The unbiased sample variance is a U-statistic for the function ƒ(x1, x2) = (x1 − x2)2/2, meaning that it is obtained by averaging a 2-sample statistic over 2-element subsets of the population.

Read more about this topic:  Variance

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